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On full cover property of ordered fields. (English) Zbl 0823.12006
The author defines the notion of a full cover of an interval \([a,b]\) in an ordered field \(A\) as follows: A class \(S\) of intervals \(I\subseteq [a,b]\) is said to be a full cover of \([a, b]\) if \(\forall x\in [a,b]\) \(\exists \delta (x)> 0_ A\) in \(A\) such that \(S\) contains each closed interval \(I\subseteq [a,b]\) of length less than \(\delta (x)\) and containing \(x\). Moreover, he defines that an ordered field \(A\) has the full cover property (FC-property) provided that for each \(a,b\in A\), \(a<b\), the following statement holds: If \(S\) is a full cover of \([a,b]\), then there exists a partition \(a_ 0= x_ 0< x_ 1< \cdots < x_ m= b\), \(x_ j\in A\), such that \(| x_{k-1}, x_ k|\in S\), \(k=1,\dots, m\). Then he proves that an Archimedean ordered field \(A\) is complete iff it has the FC-property. He gives some applications of this property.
MSC:
12J15 Ordered fields
12J10 Valued fields
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References:
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